Exercise 12, § 2

(a) For a product $\prod_{\lambda \in L} E_\lambda$ of A-modules to be injective (Exercise 11), it is necessary and sufficient that each of the $E_\lambda$ be injective.

(b) Let K be a commutative field, L an infinite set and A the product ring $K^L$. Show that A is an injective A-module (use (a)) but that the ideal $a$ of A the direct sum of the factors of $K^L$ (canonically identified with ideals of A) is not an injective A-module.